In four-dimensional asymptotically flat spacetimes, an infinite tower of soft graviton modes generates the symmetry algebra of w(1+infinity) at tree level.  This algebra is a universal feature of tree-level gravitational scattering with minimally-coupled gravitational interactions.  In the first part of the talk, I will review this construction and identify an explicit form of the w(1+infinity) symmetry generators involving integrals of the stress tensor along null infinity.  In the second part, I will discuss an independent derivation of the w(1+infinity) algebra in the light-ray operator algebra of generic 4D conformal field theories.  This talk is based on work in progress with Monica Pate.